If a matrix A is symmetric as well as anti-symmetric, then which one of the following is correct ?

Options

A. A is a diagonal matrix
B. A is a null matrix
C. A is a unit matrix
D. A is a triangular matrix

Explanation

To solve the problem, we need to analyze the properties of symmetric and anti-symmetric matrices.

### Step 1: Understand the Definitions
A matrix \( A \) is said to be symmetric if:
\[ A^T = A \]
A matrix \( A \) is said to be anti-symmetric (or skew-symmetric) if:
\[ A^T = -A \]

### Step 2: Set Up the Equations
Given that \( A \) is both symmetric and anti-symmetric, we can write:
1. From symmetry: \( A^T = A \)
2. From anti-symmetry: \( A^T = -A \)

### Step 3: Equate the Two Expressions
Since both expressions represent \( A^T \), we can set them equal to each other:
\[ A = -A \]

### Step 4: Solve for \( A \)
Adding \( A \) to both sides gives:
\[ A + A = 0 \]
This simplifies to:
\[ 2A = 0 \]
Dividing both sides by 2 results in:
\[ A = 0 \]

### Conclusion
Thus, if a matrix \( A \) is both symmetric and anti-symmetric, it must be the zero matrix (null matrix).

### Final Answer
The correct conclusion is that \( A \) is the null matrix.

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